Strassen-type laws for independent random walks

Let (Xij) be a double sequence of independent, identically distributed random variables, with mean zero and variance one, whose moment generating function is finite in a neighbourhood of the origin. Let Si(t) be the partial sum process constructed from Xi. We consider the sets Fn = {γ−1nSi(n.):i ⩽ an}, where an is a nondecreasing sequence of integers and γn is a suitable normalizing sequence. We prove a strong approximation result that in particular implies a Strassen-type law if an grows slower than exponential. If an grows at an exponential rate, we prove another Strassen-type result.